1. Field of the Invention
The present invention generally relates to optimization of financial portfolios. More specifically, a sequence of carefully constructed compact linear programs is constructed, which converges to the optimal solution. The procedure can also be applied to solve stochastic mixed-integer linear programming problems.
2. Description of the Related Art
Many businesses have to make decisions about how to allocate limited resources to a group of “candidates” such that the resulting portfolio ensures a best possible outcome for certain chosen business objectives. For example, in the investment industry, fund mangers need to decide how to allocate and/or re-allocate their funds to various financial instruments to maximize return. In project management, project managers need to decide how to allocate limited budget to projects to optimize certain performance objectives (e.g., revenue, cost, strategic impact, etc.).
The general characteristics of portfolio management problems are at least as follows:                1. Decisions are made under uncertainty. In investment portfolio management, returns of financial instruments are uncertain. In project portfolio management, future revenues of projects are uncertain.        2. Resources of various types are limited. In investment portfolio management, the amount of available funds is limited. In project portfolio management, budget and human resources are limited.        3. Risk attitude towards portfolio performance significantly affects the decision. It is typically true that greater potential returns (or future revenues) are also associated with greater risk. Depending on the decision maker's risk attitude, a risky portfolio with a higher potential return might or might not be acceptable.        4. Portfolio decisions are driven by business objectives. Investment portfolio decisions are usually driven by the desire of maximizing returns. Project portfolio decisions are often driven by multiple criteria, often including revenue, cost, strategy considerations, etc.        
One can build stochastic optimization models for portfolio selection problems, and consequently solve the models to find the optimal portfolio that leads to the best outcome. In these models, uncertainty is characterized by random variables, and decision-markers' risk attitudes are incorporated by imposing bounds on risk measures.
The best known method for stochastic programs with CVaR (Conditional Value at Risk) risk measure (discussed in more detail relative to Formulation (1) below) consists of the following steps:
First, generate samples (using Monte Carlo simulation or other methods) for random variables;
Second, use the samples obtained to re-formulate the original problem as a linear program;
Third, solve the linear program to get a solution. For more details of this approach, see the article “Optimization of Conditional Value-at-Risk” by R. T. Rockafellar, et al., in the spring, 2000, Journal of Risk, Vol. 2, No. 3.
The second step of reformulation introduces a large number of auxiliary variables and constraints, leading to a linear program that is significantly larger than the original problem. Indeed, the number of auxiliary variables and constraints introduced is proportional to the number of samples obtained in the first step. This causes the method to break down for even modest-size original problems.
More generally, many financial portfolio optimization problems can be formulated as stochastic programs. Such problems typically have several risk constraints. The best known solution leads to a prohibitively large linear program.
Thus, a need exists for improving the efficiency of solving risk-constrained stochastic programs with CVaR risk measures.